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Real Analysis, differentiation

  • Thread starter gaborfk
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  • #1
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Solved: Real Analysis, differentiation

Homework Statement


If g is differentiable and g(x+y)=g(x)(g(y) find g(0) and show g'(x)=g'(0)g(x)

The Attempt at a Solution


I solved g(0)=1

and

I got as far as

[tex]
g'(x)=\lim_{\substack{x\rightarrow 0}}g(x) \frac{g(h)-1}{h}
[/tex]

but now I am stuck.

Thank you in advance
 
Last edited:

Answers and Replies

  • #2
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Never mind... Just did g'(0) and plug in...
 
  • #3
HallsofIvy
Science Advisor
Homework Helper
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Your formula for g' is incorrect- though it may be a typo.
[tex]g'(x)= \lim_{\substack{h\rightarrow 0}}g(x)\frac{g(h)- 1}{h}[/tex]
where the limit is taken as h goes to 0, not x. Since "g(x)" does not depend on h, you can factor that out:
[tex]g'(x)= g(x)\left(\lim_{\substack{h\rightarrow 0}}\frac{g(h)-1}{h}\right)[/tex]
and you should be able to see that the limit is simply the definition of g'(0).
 

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